IL NUOVO CIMENTO VOL. 108 B, N. 9 Settembre 1993

Wave-Particle Duality and the de Broglie Relation (*).

D. K. Ross

Department of Physics and Astronomy, Iowa State University - Ames, IA 50011

(ricevuto il 26 Aprile 1993; approvato il 16 Giugno 1993)

Summary. - - We show quite generally, using the classification of the relevant fiber bundle, that a field which behaves like a semi-classical periodic wave and also has pointlike interactions is characterized by n inequivalent bundles where n is an integer. We also show that the de Broglie relation arises from the bundle structure in much the same way as the Dirac quantization condition arises in the theory of magnetic monopoles.

PACS 03.65 - Quantum theory; quantum mechanics.

Wave-particle duality is built fundamentally into the conceptual foundations of quantum mechanics. The fundamental building blocks of nature such as photons, electrons, and quarks behave as point particles in their interactions and yet are des- cribed by wave functions which allow for wavelike interference. In a two-slit ap- paratus with a very low counting rate, a single electron can even interfere with itself.

Another peculiar aspect of quantum mechanics, which is logically distinct from wave-particle duality, is that, when a light wave of given frequency interacts at a point, it always does so in terms of quanta. The energy and momentum that is delivered to the interaction is always a multiple of the same amount and given by the de Brogiie relation, P7 = nk, h, where n is an integer, P~ is the four-momentum, and k~ the wave vector. The de Broglie relation is semi-classical in the sense that the wave functions or amplitudes are varying slowly enough that a meaningful frequency and wavelength and hence k, can be defined. We want to show in this brief paper very generally, for situations where periodic waves of definite frequency and wavelength exist, that wave-particle duality implies the existence of quanta and that these quanta satisfy the de Broglie relation.

Fiber bundles are ideally suited to studying non-trivial topological configurations. In the case of a gauge theory the fiber bundle, locally, is the product of a base space such as four-dimensional space-time with a gauge group such as SUa. Globally, the fiber bundle need to look like a simple direct product and we can have topologically non-trivial bundles such as a magnetic monopole associated with the electromagnetic

(*) The author of this paper has agreed to not receive the proofs for correction.

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1060 D.K. ROSS

U1 gauge theory [1]. Several good treatments of fiber bundles exist [2-4], and we will not repeat the definitions here.

Let us assume that we have the simplest possible situation of an uncharged complex scalar field r interacting only gravitationally. Since no spinors are involved in this simple example, we can consider the gravitational interaction as given by a TI gauge theory over the base-space J~r4 [5], where the structure group T4 is the four-dimensional translation group and J /4 is Minkowski space. We call this principal bundle P. (If spinor sources were involved, a more appropriate gauge group would be the Poincar~ group [6, 7]. Our uncharged scalar field can be considered as belonging to a one-dimensional vector V[4]. T4 acts on V by way of a representation p. In general, p can be a vector or tensor representation and may be real or complex matrix. We will take p to be a one-dimensional complex representation. V is thus acted on by representation matrices belonging to G/(1, 5~). Consider now a one- dimensional complex vector bundle E over ~#4- E has a fiber isomorphic to 6 ~ and structure group T4, where T4 acts on the fibers of E by Gl(1, 6 J) matrices of the representation ~. Our uncharged scalar matter field r is then a section of E and is a vector with one component. The connection F in P induces a connection FE in E [4]. E is often referred to as an associated vector bundle [3].

We now want to look at the implications of wave-particle duality. To represent the pointlike aspects, we can take the base space of the fiber bundles P and E to be the four-dimensional Minkowski space-time ~/f4 with the line singularity along the particle's world line excised, rather than -#4 itself as above. Thus, the base space is - # a - {particle world line}. This is contractible to R ~ - {spatial point} and this in turn is contractible to $2. These contractions do not affect the topology of the fiber bundle nor its classification below [4]. This base space is the same as that used by Wu and Yang in their treatment of magnetic monopoles [1].

Wave properties are now brought in. We assume that we are in a semi-classical situation as above where k~ can be defined. Let us restrict our attention to a matter field r which is a periodic plane wave given by exp [ik~,x~']. This expression is Lorentz invariant since we expect physics to obey the principles of special relativity. The periodicity is important here. Also note that this is where ku enters our considerations. A more general periodic wave function of course could be made up of a superimposition of plane waves. This more complicated periodic matter field would have the periodicity needed below, but would involve more than one k~.

Now we can choose the Gl(1, 6 ~ matrices of the representation ~ of the structure group T4 to be T(b ,~) = exp[- ib~P~/A], where x ~---) x '~ = x ~ + b ~ is the translation and the momentum p~ is the generator of the translations [8]. A is a constant with dimensions of angular momentum so that the argument of the exponential is dimensionless. This is where P, enters our calculation. This one-dimensional matrix acts on the complex numbers in the fibers of E to produce new complex numbers. If exp [ik~x ~] is a cross-section of E, after a translation acting on the fibers we will clearly get a translated plane wave, exp [ik, (x ~ + b~)]. Thus, for a periodic wave, the translation in four dimensions is equivalent to multiplying the wave function by a simple phase factor exp [ia] where a - k~ b ~ . This would not, of course, be true for a matter field with a more complicated x~-dependence. Because of the periodicity of the plane wave, four-dimensional translational transformations reduce to elements of/]1. Thus, for plane waves we can consider our matter fields to be cross-sections of a new vector bundle E ' , where E' has U1 as the structure group rather than T4. The fiber is isomorphic to 6 ~ as before and the base space is ~#4. E ' is the associated vector

WAVE-PARTICLE DUALITY AND THE DE BROGLIE RELATION 1061

bundle related to P ' by wave of a representation where P ' is a principal bundle with structure group U1 and base space -#4.

We are now ready to put the particlelike and wavelike properties together. It is clear from the above that we can do this by starting with E ' which describes a periodic plane wave over ~ 4 and letting the base space ~~4 ---) ~//f4 - {particle world line} in order to include the particlelike properties. Thus the appropriate description of an entity which exhibits both particlelike and wavelike properties is the cross-section of a bundle E" which has a/ /1 structure group, a fiber isomorphic to G, and a base space -t-f4- {particle world line}. E" is the associated vector bundle related to P" by way of a representation where P" is a principal bundle with structure group //1 and base space ~/#a - {particle world line}. We emphasize that the //1 here has to do with the effects of translations on plane waves and has nothing to do with electromagnetism.

Let us now classify the above fiber bundle representing an entity exhibiting wave-particle duality and see how many inequivalent bundles we have. As mentioned above, as far as classifying the bundle representing our wave-particle is concerned, we can contract the base space to Se. Also U: is isomorphic to $1. Thus P" is an S: bundle over $2 and E" is the vector bundle associated with P" as above. If we wish to characterize the topological properties of our wave-particle fiber bundle, we can use the following classification from Steenrod[9]: ,,The equivalence classes of bundles over S n with group G are in one-to-one correspondence with equivalence classes of elements of Hn-I(G) under the operations of IIo(G). Such a correspondence is provided by gt? ---) Z(a), where a is a generator of Hn (S ~) and ~: II,~ (S n) ---* I I n _ 1 (G) is a characteristic homeomorphism of B . , In the present case we want H: ($1).//1 measures simple connectivity and/ /1 ($1) = zz, the additive group of integers. Thus our wave-particle bundle is characterized by an integer n. For each value of n, we have topologically distinct, inequivalent bundles. These bundles cannot be converted into one another. As the particle wave propagates, n is conserved, and we have some sort of indestructible ,,quanta,. Thus wave-particle duality as interpreted above leads directly to <,quanta,. We have gotten this far on very general grounds.

Let us delve more deeply into the structure of the fiber bundle to see what these -quanta- are. Our situation is mathematically very similar to electromagnetism.

There, matter wave functions have a phase factor exp [ie/hc ~ A~ dx ~] [1]. For us,

T4---)//1 as above and the corresponding phase factor can be written as

exp[(ik'~/Ak~) I H,~ dx~].

The presence of k" is strongly suggested by the original form exp [ik, x'] of our plane wave. k" plays the role of the <,electric charge>> and k~'H~,~ the role of the charge times the vector potential or connection. We really only have a single //1 present, not four, and end up with a single quantization condition below. A, which is a constant with dimensions of angular momentum, has been introduced in the phase factor in correspondence with Planck's constant in the electromagnetic case. Finally, we show below that natural units for H~ are momentum/length, so k 2 is introduced in our phase factor to make the argument of the exponential dimensionless. If we have a gauge transformation of our matter fields of the form

(1) ~b = exp [ia]~a

1062 D. K, ROSS

as discussed earlier, then it is easy to see from our phase factor above that our connection must transform as

(2) (H~)b = (H,~)~ + Ak~, 3x ~

Now from the classification discussion above, we have a non-trivial bundle with structure group isomorphic to $1 and base space contractible to $2, exactly as in the treatment of a magnetic monopole by Wu and Yang [1]. We saw above that k ~ plays the role of ,,electric charge,. In Wu and Yang, the magnetic monopole is represented in the base space as a point particle. Thus we expect the particlelike aspects of our particle wave to play a role analogous to the magnetic monopole in their work. The quantity that best represents these particlelike aspects in our work is the four- momentum p~. Thus p , plays a mathematical role analogous to ,,magnetic charge-.

In Wu and Yang[1], if a magnetic monopole is present, the connection A~ is non-trivial and requires two overlapping regions on $2 for its definition. They choose region Ra as

(3) 0 < 0< r:/2 +~ , 0 ~ < r

and region Rb as

(4) r:/2-- ~ < 0 ~< 7:, 0<~r

We choose these same regions and take the r component associated with the second index of H~ to be

P ' ( 1 - c o s 0 ) , (5) (H~r r sin 0

- P " (1 + cos0) (6) (H~r rsin0

in the two regions R a and Rb. The other components of H,~ are zero. It is important to emphasize at this point that (5) and (6) really represent the fact that P~ is playing the role of a point source in the base space. Because our U1 derives from T4-o/-71, the analogoue of the field tensor (curvature) associated with the potential (connection) H,~ and the analogue of the ,,magnetic field, associated with the curl of (5) and (6) have no particular relevance or physical meaning. (5) and (6) tell us how a pointlike P~ acts as a source of H~ when we have a non-trivial bundle.

We can now easily derive [1] the analogoue of Dirac's quantization condition [10] for magnetic monopoles. Putting (5) and (6) into (2) gives

( 7 ) - - -

The gauge transformation in the overlap of the two regions Sab = (Sba) -1 -- exp [ - i s ] from (1) must be single-valued so (7) can be written as

A k (8) Pr~ = n ~ ~.

WAVE-PARTICLE DUALITY AND THE DE BROGLIE RELATION 1063

A is a constant with dimensions of angular momentum which we introduced earlier. If we identify A/2 = h, where h is Planck's constant, we get the usual de Broglie relation. The n in (8) is the same n we found in classifying our bundle. The factor of 2 in (8) also appears in the Dirac quantization condition for magnetic monopoles. Thus our inequivalent bundles characterized by an integer n are really de Broglie quanta satisfying the de Broglie relation if we interpret A as above. It is remarkable that quantization and the de Broglie relation follow solely from the fact that we have a field exhibiting wave-particle duality.

I would like to thank the Physics Department of the University of Canterbury, Christchurch, New Zealand for their warm hospitality while some of this work was being done.

REFERENCES

[1] T. T. Wu and C. N. YANG: Phys. Rev. D, 12, 3845 (1975). [2] Y. CHOQUET-BRUHAT, C. DE WITT-MORETTE and M. DILLARD-BLEICK: Analysis,

Manifolds, and Physics (North-Holland, Amsterdam, 1977). [3] M. E. MAYER: Fiber Bundle Techniques in Gauge Theories (Springer, Berlin, 1977). [4] C. NAS~ and S. SEN: Topology and Geometry for Physicists (Academic Press, New York,

N.Y., 1983). [5] Y. M. CH0: Phys. Rev. D, 14, 2521 (1976). [6] T. W. B. KIBBLE: J. Math. Phys., 2, 212 (1961). [7] Y. M. CHO: Phys. Rev. D, 14, 3335 (1976). [8] W.-K. TUNG: Group Theory in Physics (World Scientific, Philadelphia, Penn., 1985). [9] i . STEENROD: The Topology of Fiber Bundles (Princeton University Press, Princeton,

N.J., 1951). [10] e. A. M. DIRAC: Proc. R. Soc. London, Set. A, 133, 60 (1931).